We study the nonhomogeneous boundary value problem for the Navier-Stokes equations of steady motion of a viscous incompressible fluid in arbitrary bounded multiply connected plane or axiallysymmetric spatial domains. We prove that this problem has a solution under the sole necessary condition of zero total flux through the boundary. The problem was formulated by Jean Leray 80 years ago. The proof of the main result uses Bernoulli's law for a weak solution to the Euler equations.

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We study the nonhomogeneous boundary value problem for Navier-Stokes equations of steady motion of a viscous incompressible fluid in a two-dimensional bounded multiply connected domain Ω = Ω 1 \ Ω 2 , Ω 2 ⊂ Ω 1 . We prove that this problem has a solution if the flux F of the boundary datum through ∂Ω 2 is nonnegative (

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We study a nonhomogeneous boundary-value problem for the steady-state Navier-Stokes equations in a two-dimensional exterior domain with two orthogonal symmetry axes. The existence of a solution which tends to zero uniformly at infinity is proved under suitable parity conditions on the data of the problem. The result is obtained for arbitrary values of the flux of the boundary datum.